A standard result concerning total variation distance is as follows
[1]}. If \(\mu \)  and \(\nu \) 
are probability measures on \(E\) , then
\(\Vert \mu (\cdot ) - \nu (\cdot )\Vert  = \sup _{f:E \rightarrow [0,1]} \bigg |\int f d\mu - \int f d\nu \bigg | \;.\) 
